Simplify this polynomial:
$3 x^3+x+2 x^3-4 x^2+14 y-2(y+x)$
A. $3 x^3-3 x^2+14(-2 y+2 x)$
B. $5 x^3-3 x^2+14 y-2 y-2 x$
C. $3 x^3-x+4 x^2+12 y$
D. $5 x^3-4 x^2-x+12 y$
Answer (2)
Distribute the -2: 3 x 3 + x + 2 x 3 − 4 x 2 + 14 y − 2 ( y + x ) = 3 x 3 + x + 2 x 3 − 4 x 2 + 14 y − 2 y − 2 x .
Combine the x 3 terms: 3 x 3 + 2 x 3 = 5 x 3 .
Combine the x terms: x − 2 x = − x .
Combine the y terms: 14 y − 2 y = 12 y . The simplified polynomial is 5 x 3 − 4 x 2 − x + 12 y .
Explanation
Understanding the Problem We are asked to simplify the polynomial 3 x 3 + x + 2 x 3 − 4 x 2 + 14 y − 2 ( y + x ) . This involves combining like terms after distributing the − 2 across the ( y + x ) term.
Distributing the -2 First, distribute the − 2 to the terms inside the parentheses:
3 x 3 + x + 2 x 3 − 4 x 2 + 14 y − 2 ( y + x ) = 3 x 3 + x + 2 x 3 − 4 x 2 + 14 y − 2 y − 2 x
Grouping Like Terms Next, we combine the like terms. We have x 3 terms, x 2 terms, x terms, and y terms. Let's group them:
( 3 x 3 + 2 x 3 ) − 4 x 2 + ( x − 2 x ) + ( 14 y − 2 y )
Combining Coefficients Now, combine the coefficients of the like terms:
For the x 3 terms: 3 + 2 = 5 , so we have 5 x 3 .
For the x 2 terms: we only have − 4 x 2 .
For the x terms: 1 − 2 = − 1 , so we have − x .
For the y terms: 14 − 2 = 12 , so we have 12 y .
Simplified Polynomial Putting it all together, we get:
5 x 3 − 4 x 2 − x + 12 y
Final Answer Comparing our simplified polynomial to the given options, we see that it matches option D.
Therefore, the simplified polynomial is 5 x 3 − 4 x 2 − x + 12 y .
Examples
Polynomial simplification is a fundamental skill in algebra and is used in many real-world applications. For instance, engineers use polynomials to model curves and surfaces, and simplifying these polynomials can make calculations easier. Imagine designing a bridge where the shape of the arch is described by a polynomial. Simplifying this polynomial allows engineers to quickly calculate stress points and ensure the bridge's stability. Similarly, in computer graphics, polynomials are used to create smooth curves and surfaces for 3D models, and simplifying these polynomials can improve rendering performance.
The simplified polynomial is 5 x 3 − 4 x 2 − x + 12 y , which matches option D. To arrive at this, we distributed the -2 and combined like terms. The final answer is thus option D: 5 x 3 − 4 x 2 − x + 12 y .
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